(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 8922, 232] NotebookOptionsPosition[ 7002, 165] NotebookOutlinePosition[ 8519, 218] CellTagsIndexPosition[ 8442, 213] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ Mathematical Modelling and Scientific Computing in the Biosciences II\ \>", "Title", CellChangeTimes->{{3.400869877645982*^9, 3.400869884970946*^9}, { 3.400869925353651*^9, 3.400869959090375*^9}}, FontSize->24], Cell[CellGroupData[{ Cell["Exercise 2 (Due: 7 December)", "Section", CellChangeTimes->{{3.400916250886026*^9, 3.400916256519915*^9}, { 3.4014755231781*^9, 3.401475540317594*^9}, 3.401520941983456*^9, { 3.401522840631402*^9, 3.401522880686961*^9}, {3.404471378789228*^9, 3.4044713884199343`*^9}}, FontSize->24], Cell["\<\ The Chay-Keizer ODE model for pancreatic beta cells with ER is given in \ lecture and on course website. \ \>", "Text", CellChangeTimes->{{3.401475559671969*^9, 3.401475582451554*^9}, 3.401520941984051*^9, {3.401522921836328*^9, 3.401522972167018*^9}, { 3.404471413346736*^9, 3.404471551966792*^9}, {3.404471786824222*^9, 3.404471800970594*^9}}, FontSize->16], Cell[CellGroupData[{ Cell[TextData[{ "Question 1 (5 points): integrate the ODE system numerically (up to ", Cell[BoxData[ FormBox[ RowBox[{"time", "=", RowBox[{"1.5", "*", SuperscriptBox["10", "5"]}]}], TraditionalForm]]], "), using a stiff method (e.g., ode23s); you should observe spiking \ behavior. Plot all 4 state variables with respect to time." }], "Item1", CellChangeTimes->{{3.401476898555368*^9, 3.401476965712787*^9}, { 3.40147707978175*^9, 3.401477124531934*^9}, 3.401520942002151*^9, { 3.401521456001343*^9, 3.401521457255784*^9}, {3.401521663083643*^9, 3.401521666323462*^9}, {3.401521859732197*^9, 3.401521860320522*^9}, { 3.401521894923743*^9, 3.401522037871579*^9}, {3.401522517052127*^9, 3.401522569104853*^9}, 3.401524162242133*^9, {3.4044715643952847`*^9, 3.404471585285288*^9}, {3.404471615523329*^9, 3.404471671337144*^9}, { 3.4045391046305733`*^9, 3.404539119858511*^9}, {3.404539334628447*^9, 3.404539335014471*^9}, {3.4056657976883583`*^9, 3.405665858977833*^9}}, FontSize->14], Cell["\<\ Question 2 (3 points): as was discussed in lecture, in spiking behavior there \ is an underlying fast system controlled by a slow system. From the plots of \ Q1, what would be reasonable as the slow variable? Explain. \ \>", "Item1", CellChangeTimes->{{3.401476898555368*^9, 3.401476965712787*^9}, { 3.40147707978175*^9, 3.401477124531934*^9}, 3.401520942002151*^9, { 3.401521456001343*^9, 3.401521457255784*^9}, {3.401521663083643*^9, 3.401521666323462*^9}, {3.401521859732197*^9, 3.401521860320522*^9}, { 3.401522282635317*^9, 3.401522283472218*^9}, {3.401522543649347*^9, 3.401522544273459*^9}, {3.401522575486064*^9, 3.401522575773154*^9}, { 3.404471687176593*^9, 3.404471747044262*^9}, {3.404471818069854*^9, 3.4044718667800303`*^9}}, FontSize->14], Cell["\<\ Question 4 (15 points): Using the solution to Q2, plot the bifurcation \ diagram using the slow variable as the bifurcation parameter. You should see \ bistability as well a Hopf point (hint: you may need to continue to negative \ \"parameter\" values to find the Hopf point). Do continuation of limit cycles \ from the Hopf point. Plot both the voltage V, amd [Ca]_i , as well as the \ period of oscillations for the limit cycle. \ \>", "Item1", CellChangeTimes->{{3.401476898555368*^9, 3.401476965712787*^9}, { 3.40147707978175*^9, 3.401477124531934*^9}, 3.401520942002151*^9, { 3.401521456001343*^9, 3.401521457255784*^9}, {3.401521663083643*^9, 3.401521719310853*^9}, {3.401522289718974*^9, 3.401522357310511*^9}, { 3.401522475769002*^9, 3.401522546458574*^9}, {3.401522580589768*^9, 3.401522615188992*^9}, {3.404471899111684*^9, 3.4044721377577343`*^9}, 3.404476191529643*^9}, FontSize->14], Cell["\<\ Question 5 (2 points): what effect should varying \[Sigma]have on the burst \ period?\ \>", "Item1", CellChangeTimes->{{3.401476898555368*^9, 3.401476965712787*^9}, { 3.40147707978175*^9, 3.401477124531934*^9}, 3.401520942002151*^9, { 3.401521456001343*^9, 3.401521457255784*^9}, {3.401521663083643*^9, 3.401521719310853*^9}, {3.401522289718974*^9, 3.401522357310511*^9}, { 3.401522475769002*^9, 3.401522546458574*^9}, {3.401522580589768*^9, 3.401522615188992*^9}, {3.404471899111684*^9, 3.4044721377577343`*^9}, 3.404476191529643*^9, {3.404476396790563*^9, 3.404476564102399*^9}, { 3.404476625163846*^9, 3.404476675067226*^9}, {3.404539262527714*^9, 3.404539307173925*^9}}, FontSize->14], Cell["\<\ Question 6 (5 points): the geometry of the bifurcation diagram captures \ behavior of the time-series solution. We see in the time series that spiking \ phase starts when [Ca]_ER reaches below a certain critical value. 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